E1etf.cin

// e1etf.cin is routine that evaluates tetration to Henryk base η=exp(1/e).

// Call it with capitals: z_type E1ETF(z_type z)

// z_type should be defined as, for example, complex<double>

z_type e1etf(z_type z){ int n,N; z_type c,f;
                        z+=2.798248154231454;
                        z_type t=log(z); z_type u=-1./(3.*z); 
z_type s[16];
s[0]= 1.;
s[1]= t;
s[2]= t*(t- 1. )+ .5;
s[3]= t*(t*(t- 5/ 2.)+ 5/ 2.)- 7/10.;
s[4]= t*(t*(t*(t- 13/ 3.)+ 45/ 6.)- 53/10.)+ 67/60.;
s[5]= t*(t*(t*(t*(t- 77/12.)+ 101/ 6.)- 83/ 4.)+ 653/60.)- 2701/1680.;
s[6]= t*(t*(t*(t*(t*(t- 87/10.)+ 95/ 3.)- 175/ 3.)+ 267/ 5.)-17245/ 840.)+ 92461/42000.;
s[7]= t*(t*(t*(t*(t*(t*(t-223/20.)+1591/30.)- 535/ 4.)+ 5488/30.)-30061/ 240.)+ 503159/14000.)-348617/84000.;
s[8]=t*(t*(t*(t*(t*(t*(t*(t-481/35.)+2947/36.)-8011/30.)+29977/60.)- 9305/ 18.)+11298583/42000.)-580789/8400.)+4558331/352800.;
s[9]=t*(t*(t*(t*(t*(t*(t*(t*(t-4609/280.)+16659/140.)-43417/90.)+349903/300.)-199529/120.)+166822247./126000.)
       -48732671./84000.)+21806039./117600.)-12523881847./388080000.;
N=9; c=s[N]*u; 
for(n=N-1;n>0;n-=1){c+=s[n];c*=u;} c+=1.;
return M_E*(1.-(2./z)*c);
}

z_type E1ETF(z_type z){
       if(Re(z)>12) return e1etf(z);
       if(fabs(Im(z))<5.)return log(E1ETF(z+1.))*M_E;
       if(abs(z-3.)<16) return log(E1ETF(z+1.))*M_E;
               return e1etf(z);
               }

References

http://www.ams.org/journals/mcom/0000-000-00/S0025-5718-2012-02590-7/S0025-5718-2012-02590-7.pdf
http://www.ils.uec.ac.jp/~dima/PAPERS/2012e1eMcom2590.pdf
http://mizugadro.mydns.jp/PAPERS/2012e1eMcom2590.pdf H.Trappmann, D.Kouznetsov. Computation of the Two Regular Super-Exponentials to base exp(1/e). Mathematics of Computation. Math. Comp., v.81 (2012), p. 2207-2227. ISSN 1088-6842(e) ISSN 0025-5718(p)

http://mizugadro.mydns.jp/t/index.php/E1etf.cin

https://www.morebooks.de/store/ru/book/Суперфункции/isbn/978-3-659-56202-0
http://www.ils.uec.ac.jp/~dima/BOOK/202.pdf
http://mizugadro.mydns.jp/BOOK/202.pdf Д.Кузнецов. Суперфункции. Lambert Academic Press, 2014. (in Russian)